4 UNIFORMIZABLE SPACES 11 family of pseudo-distances (dx\ e , . A topology which can be defined by a family of pseudo-distances is said to be uniformizable, and a space equipped with such a topology is said to be uniformizable. It is obvious that a metrizable space is uniformizable, but the converse is false (for example, the family consisting only of the pseudo-distance J = 0 defines the chaotic topology). There exist Hausdorff topologies which are not uniformizable (Problems 3 and 4); but throughout this book (and in the majority of situations in analysis) uniformiz- able spaces are the only ones which we shall have to consider. (12.4.2) Let E be a uniformizable space whose topology is defined by a family of pseudo-distances (da)ael . Then the sets B(a; (*j\ (/v)) (resp. B'<*; (ay), (r,))) are open (resp. closed) in E. Let x e B(a; (a,-), (r,.)), and put Sj = daj(a, x). Then Sj < r,- for 1 <>j £ m, hence B(a; (a,-), (/y)) contains the set E(x; (a/), (r^ — Sj)) by virtue of the tri- angle inequality for each of the daj . This shows that E(a; (ocj), (rj)) is an open set. If x $ B'(#; (o/)» (r,-)), then there exists an index k such that 1 g k <J m and dak(a, x) = sk > rk , and hence the triangle inequality for dak shows that the set B(x\ ak,sk~~ rk) does not meet B'(<^; (o/X (o))* This shows that Br((x;(aj), (r^)) is closed. (12.4.3) In a uniformizable space, the closed neighborhoods of a point form a fundamental system of neighborhoods of the point, This follows from (12.4.2) and the fact that B'O; (a,), (0/2)) <= E(a; (a,), (ry)). (12.4.4) £^r E be a uniformizable space and let (rfa)ae j 6# a family of pseudo- distances defining the topology ofE. Then E is Hausdorff if and only if, for each pair of distinct points x, y in Ey there exists ft e I such that dfl(x, y) / 0. If E is Hausdorff, then by definition there exists a set B(JC; (ay), (ry)) not containing y, and therefore for at least one index/ we have daj(x, y)^rj> 0. Conversely, itd^x, y) = / > 0, then the open neighborhoods B(x; J8, r/2) and B(j; j^? ^/2) of x and 7, respectively, are disjoint, by virtue of the triangle inequality for dp . If (4) is a family of pseudo-distances defining a Hausdorff topology on a set E, then to say that a sequence (xn) of points of E has a point a as limit with respect to this topology means that lim da(a, xn) = Ofor all a.section of the